Mathematical Aspects of Evolving Interfaces - Gerhard Dziuk

. .

Résumé :
Interfaces are geometrical objects modelling free or moving boundaries and arise in a wide range of phase change problems in physical and biological sciences, particularly in material technology and in dynamics of patterns. Especially in the end of last century, the study of evolving interfaces in a number of applied fields becomes increasingly important, so that the possibility of describing their dynamics through suitable mathematical models became one of the most challenging and interdisciplinary problems in applied mathematics. The 2000 Madeira school reported on mathematical advances in some theoretical, modelling and numerical issues concerned with dynamics of interfaces and free boundaries. Specifically, the five courses dealt with an assessment of recent results on the optimal transportation problem, the numerical approximation of moving fronts evolving by mean curvature, the dynamics of patterns and interfaces in some reaction-diffusion systems with chemical-biological applications, evolutionary free boundary problems of parabolic type or for Navier-Stokes equations, and a variational approach to evolution problems for the Ginzburg-Landau functional.

Biographie:
Prof. Luigi Ambrosio is a Professor of Mathematical Analysis, a former student of the Scuola Normale Superiore and presently its Rector. His research interests include calculus of variations, geometric measure theory, optimal transport and analysis in metric spaces. For his scientific achievements, he has been awarded several prizes, in particular the Fermat prize in 2003, the Balzan Prize in 2019, the Riemann Prize in 2023 and the Nemmers Prize in 2024. Dr. Elia Bru? is an Associate Professor at Bocconi University, Milan, Italy. He earned his PhD from the Scuola Normale Superiore in 2020. His research interests lie in the fields of Geometric Analysis and Partial Differential Equations, with a focus on Ricci curvature, metric geometry, incompressible fluid mechanics, and passive scalars with rough velocity fields. Dr. Daniele Semola is an Assistant Professor at the University of Vienna. He was a student in Mathematics at the Scuola Normale Superiore, where he earned his PhD degree in 2020. His research interests lie at the interface between geometric analysis and analysis on metric spaces, mainly with a focus on lower curvature bounds....

Sommaire:
Preface.- 1. L. Ambrosio: Lecture Notes on Optimal Transport Problems.- 2. K. Deckelnick and G. Gziuk: Numerical Approximation of Mean Curvature Flow of Graphs and Level Sets.- 3. M. Mimura: Reaction-Diffusion Systems Arising in Biological and Chemical Systems: Application of Singular Limit Procedures.- 4. V. A. Solonnikov: Lectures on Evolution Free Boundary Problems: Classical Solutions.- 5. H. M. Soner: Variational and Dynamic Problems for the Ginzburg-Landau Functional.